There's a scene in the Oscar-winning film A Beautiful Mind in which Russell Crowe as mathematician John Nash asks his beautiful wife-to-be to gaze into the evening sky and name any shape—an umbrella, for instance—and then says he will find the shape in a constellation. He does it, and she is utterly charmed, saying, "Do it again."
UCLA mathematician Terence Tao, 32, relates to that scene, perhaps because it reminds him of his own wife, Laura, an engineer at NASA's Jet Propulsion Laboratory, but certainly because it illustrates one of his most famous theorems. Prime numbers, those that can be divided only by themselves and 1 (that is, 2, 3, 5, 7, 11, and so on), can be visualized as points on a line. But it is more illuminating to think of them as stars in the sky, scattered more or less at random over a vast canvas. Tao, together with Cambridge University mathematician Ben Green, proved that this canvas contains patterns of every conceivable shape. Somewhere there's a prime number constellation with the same shape as Nash's umbrella. Green and Tao's discovery was the mathematical sensation of 2004.
Tao's specialty is analysis, the area of math that includes calculus and differential equations. Sylvain E. Cappell, a professor at NYU's Courant Institute of Mathematical Sciences, calls him "the leading analyst of his generation." Yet the Green-Tao theorem resolved a major question in number theory, a completely separate field. It was as unexpected as a violinist suddenly winning a major piano competition. For Tao, however, wandering across disciplinary boundaries is commonplace.
For instance, one day in 2004, Emmanuel Candes, an applied mathematician at Caltech, told Tao about a problem he was working on—how to reconstruct images with the least possible information. Present-day digital cameras go about this in a most inefficient way. They record several million pixels (the basic elements of digital pictures), then use computer instructions called a compression algorithm to reduce the amount of data in the picture by 10 or 50 times. Why not design a camera that would acquire only a 50th of the data to start with?
His reaction was vintage Tao. First he told Candes the problem was unsolvable. Then a couple of minutes later, he allowed that Candes might be on to something. By the next day, Tao had solved the problem himself. Not only that, the solution marked the birth of a new field, called compressive sampling. As a result of Candes and Tao's discovery, engineers are now working on MRI scanners several times faster than today's, and even one-pixel cameras. Is Tao an engineer? No. But he saw a connection that no engineer had seen. "Whenever he touches a subject, it becomes gold very quickly," says Candes.
Tao's accomplishments have already earned him nearly every major mathematics prize. Last year, the International Mathematical Union awarded him the Fields Medal, widely considered the mathematics equivalent of a Nobel Prize. Less than a month later, Tao got a $500,000 grant from the MacArthur Foundation.
He accepts these plaudits with modesty and generally stays out of the public eye, handling most press inquiries by e-mail. Fellow mathematicians find him open and available. "Terry is as normal as it comes," says Tony Chan of the National Science Foundation, a former chairman of UCLA's mathematics department. "He can easily be lost in a crowd of UCLA freshmen."
Born in Australia, Tao taught himself arithmetic at age 2. By age 10, he was the youngest competitor ever in the International Mathematical Olympiad (a competition for high-school students), and two years later he won its gold medal. At 17, he came to America to study mathematics as a graduate student at Princeton.
There, Tao says, he finally found his peer group—people who loved math as much as he did. According to classmate Allen Knutson, what made Tao different from other child prodigies was his emotional maturity. "He was levelheaded enough that he didn't stand out," Knutson says. Says Tao's father, Billy, a pediatrician in Adelaide: "Terry was given the freedom to enjoy life." He played bridge, dabbled in music dubbing. As he was finishing up his doctorate (at the age of 20), his adviser Eli Stein saw a marked change. "He really started to take off in his last year, maybe his last semester," Stein says. "All of a sudden things started to click."
Since then, there has been no stopping him. He took his first postgraduate job at UCLA, which quickly realized it had a phenomenon on its hands. Within just four years, at age 24, Tao became the youngest full professor in UCLA's history. "Terry brings prestige to the place, not the other way around," says Chan.
In little more than a decade, Tao has also written some 140 papers. By comparison, Chan says, most pure mathematicians would be happy with three papers a year. Just as remarkable is his growing number of co-authors (50 at last count). "I have been lucky to find very good collaborators, who have taught me a lot, have introduced me to several new fields of mathematics or have shown me new insights," he says. Plus, he adds, "they are just plain fun to work with."
It's easy to put a genius on a pedestal, to ascribe his success to otherworldly talents. But mathematical research can be more like running a marathon, where sheer determination often prevails. "If he decides to prove something, he will," says Gigliola Staffilani of MIT, a sometime collaborator. Knutson says that Tao has shown him how to chip away at a large problem a little bit at a time. "I'd say I don't understand why this thing would be true. He'd say it reduces to checking these 17 cases, and they all work. I'd say that's awesome, but we obviously can't publish that. But after a while, we'd unwind it to three cases, and then we could publish."
The heart of Tao's gift may simply be his ability to let his thinking roam freely toward an unseen horizon. "Terry is unusual in how open-minded he is," says Ben Green, his collaborator on the prime-number problem. "When we started, a lot of senior mathematicians probably would have said that the idea won't work, that it was ludicrously ambitious. He was willing to try all lines of inquiry."
Willing, you might say, to trace an umbrella in the evening sky.
Dana Mackenzie has a PhD degree in mathematics from Princeton University. He writes about science and mathematics.